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Revision Notes on Unit and Dimensions

In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.

System of Units:-

(a) C.G.S (Centimeter-Grand-Second) system.

(b) F.P.S. (Foot-Pound-Second) system.

(d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit.

Dimensional Formula:-

Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity.

How to write dimensions of physical quantities:-

(a) Write the formula for that quantity, with the quantity on L.H.S. of the equation.

(b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time.

(d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively.

Characteristics of Dimensions:-

(a) Dimensions of a physical quantity are independent of the system of units.

(b) Quantities having similar dimensions can be added to or subtracted from each other.

(d) Two different physical quantities may have same dimensions.

(e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity.

PHYSICAL QUANTITY	SYMBOL		DIMENSION		MEASUREMENT UNIT		UNIT
Length	s		L		Meter		m
Mass	M		M		Kilogram		Kg
Time	t		T		Second		Sec
Electric charge	q		Q		Coulomb		C
luminous intensity	I		C		Candela		Cd
Temperature	T		K		Kelvin		^oK
Angle	q		none		Radian		None
Mechanical Physical Quantities (derived)
PHYSICAL QUANTITY		SYMBOL		DIMENSION		MEASUREMENT UNIT	UNIT

Area		A		L²		square meter	m²
Volume		V		L³		cubic meter	m³
velocity		v		L/T		meter per second	m/sec
angular velocity		w		T^-1		radians per second	1/sec
acceleration		a		LT^-2		meter per square second	m/sec²
angular acceleration		a		T^-2		radians per square second	1/sec²
Force		F		MLT^-2		Newton	Kg m/sec²
Energy		E		ML²T^-2		Joule	Kg m²/sec²
Work		W		ML²T^-2		Joule	Kg m²/sec²
Heat		Q		ML²T^-2		Joule	Kg m²/sec²
Torque		t		ML²T^-2		Newton meter	Kg m²/sec²
Power		P		ML²T^-3		watt or joule/sec	Kg m²/sec³
Density		D or ρ		ML^-³		kilogram per cubic meter	Kg/m³
pressure		P		ML^-1T^-²		Newton per square meter	Kg m^-1/sec²
impulse		J		MLT^-1		Newton second	Kg m/sec
Inertia		I		ML²		Kilogram square meter	Kg m²
luminous flux		f		C		lumen (4Pi candle for point source)	cd sr
illumination		E		CL^-²		lumen per square meter	cd sr/m²
entropy		S		ML²T^-²K^-1		joule per degree	Kg m²/sec²K
Volume rate of flow		Q		L³T^-1		cubic meter per second	m³/sec
kinematic viscosity		n		L²T^-1		square meter	m²/sec
						per second
dynamic viscosity		m		ML^-1T^-1		Newton second per square meter	Kg/m sec
specific weight		g		ML^-²T^-²		Newton per cubic meter	Kg m^-2/sec²
Electrical Physical Quantities (derived)
Electric current	I		QT^-1		Ampere		C/sec
emf, voltage, potential	E		ML²T^-²Q^-1		Volt		Kg m²/sec²C
resistance or impedance	R		ML²T^-1Q^-²		ohm		Kgm² /secC²
Electric conductivity	s		M^-²L^-²TQ²		mho		secC²/Kg m³
capacitance	C		M^-1L^-²T² Q²		Farad		sec²C²/Kgm²
inductance	L		ML²Q^-²		Henry		Kg m² /C²
Current density	J		QT^-1L^-²		ampere per square meter		C/sec m²
Charge density	r		QL^-³		coulomb per cubic meter		C/m³
magnetic flux, Magnetic induction	B		MT^-1Q^-1		weber per square meter		Kg/sec C
magnetic intensity	H		QL^-1T^-1		ampere per meter		C/m sec
magnetic vector potential	A		MLT^-1Q^-1		weber/meter		Kg m/sec C
Electric field intensity	E		MLT^-²Q^-1		volt/meter or newton/coulomb		Kg m/sec² C
Electric displacement	D		QL^-²		coulomb per square meter		C/m²
permeability	m		MLQ^-²		henry per meter		Kg m/C²
permittivity,	e		T²Q²M^-1L^-³		farad per meter		sec²C²/Kgm³
dielectric constant	K		M⁰L⁰T⁰		None		None
frequency	f or n		T^-1		Hertz		sec^-1
angular frequency	W		T^-1		radians per second		sec^-1
Wave length	l		L		Meters		M

Principle of homogeneity:-

It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.”

Types of error:-

(a) Constant errors:- An error is said to be constant error if it affects, every time, a measurement in a similar manner.

(b) Systematic errors:- Errors which come into existence by virtue of a definite rule, are called systematic errors.

(c) Random error or accidental error:- Error which takes place in a random manner and cannot be associated with a systematic cause are called random or accidental errors.

(d) Absolute error:- $\Delta x_{i} = x_{i} - \bar{x}$